OF THE RECIPROCAL OF A DETERMINANT. 



623 



(?«! - TjMjXjBg - T. 2 «.,)(?»3 - T S U S ) + 2, («h ~ T l M l)( W 2 ~ r 2 M 2) " T s( M 3 ~ %) 





Wo "~ De£Ccy 



C 2 X 2 



"3*3 W 3 ^a 



'1'2'3 



- Vl 



a 9 x 9 it., - b 9 x, 





1* 2 C 2*2 



3*3 W 3 — C 3* C 3 



where the addition of the elements of each column shows that the three-line deter- 

 minant vanishes and that the two-line determinant has x x for a factor. As a con- 

 sequence we have 



o o 



= j + N? T s (% -% x z) + *NJ^ t 2 t b(Vi j 'i + h c \ x 3 + h \ c 2 x J • X l 



2 



(m x - t 1 ?« 1 )(ot 2 - r 2 u 9 )(m 3 - T3M3) ' ./ , ot 3 -t 3 « 8 ' / t (m. 2 -T. 2 u 2 )(m 3 -T 3 u 3 ) 



and thus see that, since the first fraction on the right has for a factor t 3 without x 3 , and 

 the second has x^ without t 1 , there results 



( (m 1 - t^j 



)(m 8 - tV)(?« 3 - T3W3) 



-'■ 



as was to be proved. 



(14) Putting m-y — rn 2 = m 3 = ....= in the theorem of the preceding paragraph we 

 obtain Jacobi's case of 1829 with which we started ; and putting mj = m 2 = m 3 = . . . = 1 

 we obtain the case which is the subject of a memoir by Major MacMahon, printed in 

 the Transactions of the Royal Society of London for 1893.* 



The new per- reciprocal theorem of § 12 of course degenerates in like fashion. 



(15) A still wider generalisation, however, is possible. All that is requisite is to 

 make use of a simple fact employed by Jacobi in the memoir above noted, viz., that if 

 «i . ft> 2 , . . . be what u x , w 2 , . . . become when x x , x 2 , . . . are changed into ^ , £ 2 , . . . , 

 — in other words, if ^ , £ 2 , . . . be the values of x t , a? 2 , . . . which satisfy the equations 

 «] = w 1 , u 2 = » 2 > • • • j — then any identity connecting u x ,u 2 ,... and the quantities of 

 which u x , u 2 , . . . are functions will still remain an identity if in every instance x r be 

 changed into x r — % r and u r into u r — w r . As a consequence of this action we obtain from 

 the result of § 12 the theorem, — 



In the expansion of 



{ m i - T l( M l " «*l)} • l'»2 - T 2( M 2 " w 2)} 



the aggregate of the terms which are such that every r occurring in them is accom- 

 panied by the corresponding x — £, and vice-versa, is 



m x - o^at! - £) - b^Xy - £) - e^fo - £) 



- «2 T 2( X 2 - 4) m 2 ~ & 2 T 2( a; 2 ~ 4) ~ C 1 T <l X -2 ~ £>) 



- a 3 T 3 (a;3 - £ 8 ) - Vsto ~ 4) m 3 - <V s (a: 8 - i 9 ) 



* MacMahon, P. A., " A certain class of generating functions in the theory of numbers,'' Trans. Hoy. Soc, 

 clxxxv. pp. 111-160. 



TRANS. ROY. SOC. EDIN., VOL. XL. PART III. (NO. 25). 4 z 



